Using Arrays for Multiplying Fractions

Transcription

1 s e s s i o n A. 6 Using Arrays for Multiplying Fractions Math Focus Points Using arrays to represent multiplication of fractions Developing an algorithm for multiplying fractions Today s Plan activity Using Arrays to Multiply Fractions Discussion Multiplying Fractions Using Arrays SESSION FOLLOW-UP 5 Min Class Individuals 15 Min Class Materials Student Activity Book, pp or C6 C7, Using Arrays to Multiply Fractions Make copies. (as needed) Student Activity Book, p. 87 or C7 (completed) Student Activity Book, p. 88 or C8, Planting Vegetables Make copies. (as needed) Student Activity Book, p. 89 or C9, Hiking Trails Make copies. (as needed) Ten-Minute Math Estimation and Number Sense: Closest Estimate Write each of the following problems on the board, one at a time: 1. _ _ Give students approximately 0 seconds to look at the three possible estimates and determine which is the closest to the actual answer. Ask several students to explain how they chose an estimate, including how they thought about each of the numbers. CC56 Investigation A Multiplying and Dividing Fractions

2 1 Activity Discussion Session Follow-Up A C T I V I T Y 5 Min CLASS Using Arrays to Multiply Fractions Individuals You ve been doing some work multiplying fractions and have come up with a conjecture that seems to work to find the answer. You can multiply the denominators to get the size of the pieces, and multiply the numerators to get the number of those pieces. We re going to continue thinking about the rule, and this time we re going to use an array to multiply fractions. If necessary, do a quick review of using an array for multiplication. Ask students how an array could be used to show. Draw a sketch on the board, and label the dimensions and area. Draw a large square on the board. We re going to solve some problems about planting crops. A square like this is going to represent 1 section of land. A section of land has an area of 1 square mile. Here s the problem: Mitch owns _ of a section of farmland, and he s going to plant potatoes on 1_ of his land. What fraction of a section will be planted with Mitch s potatoes? Let s think about how we could use an array to show 1_ _. First, how do we show how much of this section of land Mitch owns? As students respond, divide the square into fourths using vertical lines. Shade _. How can we show 1_ of Mitch s land? Divide the _ in 1_ using a dotted horizontal line. I m going to put in some stripes to show this is the part of Mitch s land where he is planting potatoes. Session A.6 Using Arrays for Multiplying Fractions CC57

3 1 Activity Discussion Session Follow-Up The question we re answering is what fraction of a section is planted with potatoes. What s the answer? How do you know? Students might say: I had to imagine the dotted line that shows 1_ going all the way across, and then I could see the whole square is divided into eighths. Three of them are shaded and have potatoes in them, so it s _ 8. I used our rule of multiplying numerators and denominators. 1_ _. 1, and 8, so it s _ 8. Let s figure out how to label the dimensions of the array. First, be sure students understand that the dimension of each side of the square is 1. (The square represents 1 square mile, so the dimensions of the sides must be 1.) Next, point to the top half of the left side. What is this dimension? What fraction of the land is it showing? I ll label this dimension 1_. Point to the shaded portion of the top side of the square. This entire dimension is _. What is the dimension of each part? I ll label each of these parts 1_. Look at this first little rectangle. Its dimensions are 1_ by 1_. What s its area? I ll label this rectangle 1_. If we look at the area of 8 the land planted with potatoes, we see that 1_ by _ is _ 8. 1 _ 1 _ 1 _ 1 _ 1 8 _ 1 _ CC58 Investigation A Multiplying and Dividing Fractions

4 1 Activity We ve shown that 1_ _ = _8. Think about the original problem. Where do we see the _ of a section of land that Mitch owns? And where do we see the 1_ of his land where he planted potatoes? Ask students to look at Problem 1 on Student Activity Book page 86 or C6, and solve it together. Repeat the process as described above, including labeling the array and making the connections between the problem and the dimensions and area shown. If some 6 = 1_. students say the answer is 1_, acknowledge that 1 _ Using Arrays to Multiply Fractions (page 1 of ) Solve the problems. Use the squares to show your work. 1. Alicia owns _ of a section of land. She plants pumpkins on _ of her land. What with Alicia s pumpkins?. Olivia owns _ of a section of land. She plants tomatoes on of her land. What with Olivia s tomatoes? Name Pearson Education 5 Session Follow-Up. Zachary owns _ of a section of land. He plants corn on _ of his land. What with Zachary s corn? _ Discussion 86 Unit Session A.6 Student Activity Book, Unit, p. 86; Resource Masters, C6 INV1_SE05_U.indd 86 Name 6/8/11 1:5 PM Using Arrays to Multiply Fractions (page of ) Solve the problems. Use the squares to show your work.. Yumiko owns _56 of a section of land. She plants peanuts on _ of her land. What with Yumiko s peanuts? Complete the remaining problems on Student Activity Book pages or C6 C7. Think about how the conjecture for multiplying denominators and numerators works with these problems. 5. Walter owns of a section of land. He plants lettuce on _87 of his land. What with Walter s lettuce? Ongoing Assessment: Observing Students at Work 6. Terrence owns _ of a section of land. He plants peas on _ of his land. What with Terrence s peas? Pearson Education 5 Students use arrays to represent multiplication problems involving two fractions, and they solve the problems. Session A.6 Can students draw an array that correctly matches the 87 Student Activity Book, Unit, p. 87; Resource Masters, C7 INV1_SE05_U.indd 87 problem? Are they able to keep track of each part of the problem first shading in the fractional amount of land, then striping the fractional part that is planted? Unit 6/8/11 1:7 PM How do students solve the problem? Do they use the array and count the total amount of pieces in the whole square and then count the shaded and striped pieces? Do they multiply the numerators and the denominators? Session A.6 INV1_TE05_U0_SA.6.indd 59 Using Arrays for Multiplying Fractions CC59 6/10/11 1: PM

5 1 Activity Discussion Session Follow-Up differentiation: Supporting the Range of Learners Some students may find it challenging to make arrays that match the problems. Ask these students about each part of the problem and relate each part to one of the steps of creating an array. Encourage these students to draw the dotted lines all the way across the square, to show how many equal pieces are in the square. Students who can easily represent and solve a fraction multiplication problem using an array can make up their own problems to solve. D i s c u s s i o n Multiplying Fractions Using Arrays Math Focus Points for Discussion Using arrays to represent multiplication of fractions Developing an algorithm for multiplying fractions 15 MIN Class Before the discussion begins, draw the array shown below for Problem on Student Activity Book page 87 or C7 ( _ 5_ ) on 6 the board. 1 _ 1 _ 5 6 _ 1 _ 1 _ 1 _ 1 _ CC60 Investigation A Multiplying and Dividing Fractions

6 1 Activity Students might say: I drew an array like your drawing, and then 1 figured out that one little rectangle is 18 of the whole square. There were 5 little rectangles shaded and striped in each of the top rows, so that was rectangles. The answer is 18 I made an array because you said we had to. But I just used the rule. _ _56. 5 is , 6 is 18, so it s 18 Session Follow-Up Name Planting Vegetables note Students use arrays to solve problems involving multiplying a fraction by a fraction. Solve the problems. Use the squares to show your work. 1. Samantha owns of a section of land. She plants squash on of her land. What with Samantha s squash?. Cecilia owns of a section of land. She plants tomatoes on _1 of her land. What with Cecilia s tomatoes?. Renaldo owns _ of a section of land. He plants corn on of his land. What with Renaldo s corn? 88 Pearson Education 5 Most of you drew an array that looked like this, although you may have labeled it slightly differently. Who is willing to tell us how you found the answer to _ 5_6? Discussion Unit Session A.6 Student Activity Book, Unit, p. 88; Resource Masters, C8 INV1_SE05_U.indd 88 [Hana] brings up an interesting point. Many of you just used the rule. I m curious as to whether or not you see any connection between the rule and the array we have for _ 5_6. In the last session, someone said that if you multiply the denominators, that s telling you how many total pieces you re going to have in your answer. And that if you multiply the numerators, that tells you how many of those pieces you re concerned about. What do people think? Where do we see 6 in the array? Where do we see 5? Talk to a neighbor. Name 6/8/11 1:8 PM Hiking Trails Solve the following problems, using fraction bars as necessary. Write an equation for each problem. note Students solve problems involving multiplying a whole number by a fraction or a mixed number. Mitch went on a hiking trip. On the first day, he hiked the 6-mile-long Wildflower Trail. 1. On the second day, Mitch hiked the Lake Point Trail. The Lake Point Trail is _ as long as the Wildflower Trail. How long is the Lake Point Trail?. On the third day, Mitch hiked the Pine Cone Trail. The Pine Cone Trail is 16 times as long as the Wildflower Trail. How long is the Pine Cone Trail? Give students a few moments to discuss this and then collect student responses. Point out connections between their ideas and the array.. On the fourth day, Mitch hiked the Big Timbers Trail. The Big Timbers Trail is 1 times as long as the Wildflower Trail. How long is the Big Timbers Trail? Pearson Education 5. On the fifth day, Mitch hiked the Golden Sands Trail. The Golden Sands Trail is _ times as long as the Wildflower Trail. How long is the Golden Sands Trail? SESSION FOLLOW-UP Session A.6 Unit 89 Student Activity Book, Unit, p. 89; Resource Masters, C9 INV1_SE05_U.indd 89 5/18/11 7:5 PM : For reinforcement of this unit s content, have students complete Student Activity Book page 88 or C8. For ongoing review, have students complete Student Activity Book page 89 or C9. Session A.6 INV1_TE05_U0_SA.6.indd 61 Using Arrays for Multiplying Fractions CC61 6/10/11 :0 PM

7 Name Using Arrays to Multiply Fractions (page 1 of ) Solve the problems. Use the squares to show your work. 1. Alicia owns _ of a section of land. She plants pumpkins on _ of her land. What with Alicia s pumpkins?. Olivia owns _ of a section of land. She plants tomatoes on 1_ of her land. What with Olivia s tomatoes?. Zachary owns _ of a section of land. He plants corn on _ of his land. What with Zachary s corn? Unit Session A.6 C6 Pearson Education, Inc., or its affiliates. All Rights Reserved. 5

8 Name Using Arrays to Multiply Fractions (page of ) Solve the problems. Use the squares to show your work.. Yumiko owns 5_ of a section of land. She 6 plants peanuts on _ of her land. What with Yumiko s peanuts? 5. Walter owns 1_ of a section of land. He plants lettuce on 7_ of his land. What 8 with Walter s lettuce? 6. Terrence owns _ of a section of land. He plants peas on _ of his land. What with Terrence s peas? Unit Session A.6 C7 Pearson Education, Inc., or its affiliates. All Rights Reserved. 5

9 Name Planting Vegetables Solve the problems. Use the squares to show your work. note Students use arrays to solve problems involving multiplying a fraction by a fraction. 1. Samantha owns 1_ of a section of land. She plants squash on 1_ of her land. What with Samantha s squash?. Cecilia owns 1_ of a section of land. She plants tomatoes on 1_ of her land. What with Cecilia s tomatoes?. Renaldo owns _ of a section of land. He plants corn on 1_ of his land. What with Renaldo s corn? Unit Session A.6 C8 Pearson Education, Inc., or its affiliates. All Rights Reserved. 5

10 Name Hiking Trails Solve the following problems, using fraction bars as necessary. Write an equation for each problem. note Students solve problems involving multiplying a whole number by a fraction or a mixed number. Mitch went on a hiking trip. On the first day, he hiked the 6-mile-long Wildflower Trail. 1. On the second day, Mitch hiked the Lake Point Trail. The Lake Point Trail is _ as long as the Wildflower Trail. How long is the Lake Point Trail?. On the third day, Mitch hiked the Pine Cone Trail. The Pine Cone Trail is 1 1_ times as long as the Wildflower 6 Trail. How long is the Pine Cone Trail?. On the fourth day, Mitch hiked the Big Timbers Trail. The Big Timbers Trail is 1 1_ times as long as the Wildflower Trail. How long is the Big Timbers Trail?. On the fifth day, Mitch hiked the Golden Sands Trail. The Golden Sands Trail is _ times as long as the Wildflower Trail. How long is the Golden Sands Trail? Unit Session A.6 C9 Pearson Education, Inc., or its affiliates. All Rights Reserved. 5